Subgroups — #35423301

Order 21248.

638 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₁₆₆	C₂	1	2	yes	maximal
2	C₂×#19127831	C₂	1	1	yes	maximal, characteristic
3	#24908238	C₂	1	2	yes	maximal
4	#24908237	C₂	1	4	yes	maximal
5	#24907636	C₂	1	2	yes	maximal
6	#24907635	C₂	1	1	yes	maximal, characteristic
7	#24907371	C₂	1	2	yes	maximal
8	C₄×#108×C₈₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
10	#19127831	—	2	2	no	—
11	#19127831	C₄	1	4	yes	—
12	#19127831	C₄	1	4	yes	—
13	#19127829	—	2	4	no	—
14	C₈×Dic₁₆₆	—	2	2	no	—
15	C₂×#14581728	C₂²	1	2	yes	—
16	C₂×#14581728	C₂²	1	2	yes	—
17	C₂×#14581727	C₂²	1	4	yes	—
18	#19127725	C₂²	1	2	yes	—
19	#19127724	C₂²	1	4	yes	—
20	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂×Dic₈₃⋊C₈	C₂²	1	2	yes	—
22	M₁₆×Dic₈₃	C₂²	1	4	yes	—
23	C₂×#14581702	C₂²	1	2	yes	—
24	#19127465	C₂²	1	4	yes	—
25	C₂×#108×C₈₃	C₂²	1	1	yes	characteristic
26	C₂×#108×C₈₃	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₈₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
29	#341×C₈₃	C₂²	1	2	yes	—
30	C₂²×Dic₁₆₆	Q₈	1	1	yes	characteristic
31	C₂×#13050574	C₂³	1	4	yes	—
32	C₂²×C₈₃⋊C₈	C₂³	1	2	yes	—
33	C₄×Dic₁₆₆	C₂×C₄	1	2	yes	—
34	C₄×Dic₁₆₆	C₂×C₄	1	2	yes	—
35	C₄×Dic₁₆₆	D₄	1	2	yes	—
36	C₄×Dic₁₆₆	D₄	1	2	yes	—
37	C₂×C₄⋊Dic₈₃	Q₈	1	1	yes	characteristic
38	C₂×Dic₈₃⋊C₄	Q₈	1	2	yes	—
39	C₂×C₄×Dic₈₃	C₂³	1	2	yes	—
40	#14581728	—	2	4	no	—
41	#14581728	—	4	2	no	—
42	#14581727	—	4	4	no	—
43	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
44	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
45	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
46	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
47	Dic₈₃⋊C₈	—	2	4	no	—
48	Dic₈₃⋊C₈	—	2	4	no	—
49	Dic₈₃⋊C₈	C₂×C₄	1	8	yes	—
50	#14581702	—	2	4	no	—
51	#14581702	C₂×C₄	1	8	yes	—
52	C₈×Dic₈₃	—	2	4	no	—
53	C₂×M₁₆×C₈₃	C₂³	1	2	yes	—
54	C₂×M₁₆×C₈₃	C₂³	1	2	yes	—
55	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
58	#108×C₈₃	—	2	2	no	—
59	#108×C₈₃	Q₈	1	4	yes	complemented
60	#97×C₈₃	—	2	2	no	—
61	C₄×C₈×C₈₃	—	2	2	no	—
62	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
64	C₂×Dic₁₆₆	—	4	2	no	—
65	C₂×Dic₁₆₆	C₄⋊C₄	1	2	yes	—
66	C₂×Dic₁₆₆	C₄⋊C₄	1	2	yes	—
67	C₂²×Dic₈₃	C₂×Q₈	1	2	yes	—
68	C₂²×Dic₈₃	C₂×Q₈	1	2	yes	—
69	#13050574	—	2	4	no	—
70	#13050574	—	2	4	no	—
71	C₂×C₈₃⋊C₈	—	2	2	no	—
72	C₂×C₈₃⋊C₈	—	2	2	no	—
73	C₂×C₈₃⋊C₈	—	2	4	no	—
74	C₂×C₈₃⋊C₈	C₂²×C₄	1	4	yes	—
75	C₄⋊Dic₈₃	C₄⋊C₄	1	2	yes	—
76	C₄⋊Dic₈₃	C₄⋊C₄	1	2	yes	—
77	Dic₈₃⋊C₄	C₄⋊C₄	1	4	yes	—
78	Dic₈₃⋊C₄	C₄⋊C₄	1	4	yes	—
79	C₄×Dic₈₃	C₂²×C₄	1	4	yes	—
80	C₄×Dic₈₃	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
84	M₁₆×C₈₃	—	2	2	no	—
85	M₁₆×C₈₃	—	2	2	no	—
86	M₁₆×C₈₃	—	2	2	no	—
87	M₁₆×C₈₃	C₂×Q₈	1	4	yes	—
88	C₂×C₈×C₈₃	—	2	2	no	—
89	C₂×C₈×C₈₃	—	2	1	no	—
90	C₂×C₈×C₈₃	—	2	1	no	—
91	C₂×C₈×C₈₃	—	2	2	no	—
92	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₈₃	#41	1	2	yes	—
96	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
100	C₂³×C₈₃	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₈₃	—	2	2	no	—
102	C₂×C₄×C₈₃	—	4	1	no	—
103	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂×C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
113	C₂×Dic₈₃	—	4	2	no	—
114	C₂×Dic₈₃	—	4	2	no	—
115	C₂×Dic₈₃	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×Dic₈₃	C₂×C₄⋊C₄	1	4	yes	—
117	Dic₁₆₆	—	2	4	no	—
118	Dic₁₆₆	—	4	2	no	—
119	C₈×C₈₃	—	2	2	no	—
120	C₈×C₈₃	—	2	4	no	—
121	C₈×C₈₃	—	4	2	no	—
122	C₈×C₈₃	C₄×Q₈	1	4	yes	—
123	C₈₃⋊C₈	—	2	4	no	—
124	C₈₃⋊C₈	—	4	4	no	—
125	C₂²×C₈₃	—	2	1	no	—
126	C₂²×C₈₃	—	2	1	no	—
127	C₂²×C₈₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₈₃	#557	1	1	yes	characteristic
129	C₂²×C₈₃	Q₈²	1	1	yes	characteristic
130	C₄×C₈₃	—	2	1	no	—
131	C₄×C₈₃	—	2	1	no	—
132	C₄×C₈₃	—	2	1	no	—
133	C₄×C₈₃	#430	1	2	yes	—
134	C₄×C₈₃	C₂×C₄×Q₈	1	1	yes	characteristic
135	C₄×C₈₃	#520	1	1	yes	characteristic
136	C₄×C₈₃	D₄×Q₈	1	1	yes	characteristic
137	C₄×C₈₃	#550	1	1	yes	characteristic
138	Dic₈₃	—	2	4	no	—
139	Dic₈₃	—	2	4	no	—
140	#22733	—	83	1	no	maximal, Sylow
141	C₂×C₈₃	—	2	1	no	—
142	C₂×C₈₃	—	2	1	no	—
143	C₂×C₈₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₈₃	C₂×#430	1	1	yes	characteristic
145	C₂×C₈₃	#2916	1	1	yes	characteristic
146	Q₈×M₁₆	—	83	2	no	—
147	C₂×#446	—	83	1	no	—
148	#1950	—	83	2	no	—
149	#1950	—	83	4	no	—
150	#1849	—	83	2	no	—
151	#1848	—	83	1	no	—
152	#1778	—	83	2	no	—
153	C₄×#108	D₈₃	1	1	yes	characteristic
154	C₈₃	#22733	1	1	yes	characteristic, complemented, Sylow
155	C₂×C₄×Q₈	—	83	1	no	—
156	#446	—	83	4	no	—
157	#446	—	83	4	no	—
158	#446	—	166	2	no	—
159	#446	—	166	4	no	—
160	C₈×Q₈	—	166	2	no	—
161	C₂×#108	—	83	4	no	—
162	C₂×#108	—	83	2	no	—
163	C₂×#108	—	83	2	no	—
164	C₂×#108	C₂×D₈₃	1	1	yes	characteristic
165	C₂×#108	C₂×D₈₃	1	1	yes	characteristic
166	#423	—	83	4	no	—
167	#423	—	83	2	no	—
168	C₂×C₄⋊C₈	—	83	2	no	—
169	C₂×C₄⋊C₈	—	83	1	no	—
170	C₄×M₁₆	—	83	4	no	—
171	C₄×M₁₆	C₂×D₈₃	1	2	yes	—
172	C₂×#97	—	83	2	no	—
173	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
174	#341	—	83	4	no	—
175	#341	C₂×D₈₃	1	2	yes	—
176	C₂²×Q₈	—	83	1	no	—
177	C₂×M₁₆	—	83	4	no	—
178	C₂×M₁₆	C₂²×D₈₃	1	2	yes	—
179	C₂×M₁₆	C₂²×D₈₃	1	2	yes	—
180	C₂²×C₈	—	83	2	no	—
181	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
183	C₄×Q₈	—	83	2	no	—
184	C₄×Q₈	—	83	2	no	—
185	C₄×Q₈	—	83	2	no	—
186	C₄×Q₈	—	83	2	no	—
187	C₂×C₄⋊C₄	—	83	2	no	—
188	C₂×C₄⋊C₄	—	83	1	no	—
189	C₂×C₄²	—	83	2	no	—
190	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
191	#108	—	2	2	no	—
192	#108	—	332	2	no	—
193	#108	—	332	4	no	—
194	#108	—	166	4	no	—
195	#108	Dic₁₆₆	1	4	yes	complemented
196	C₄⋊C₈	—	83	8	no	—
197	C₄⋊C₈	—	83	2	no	—
198	C₄⋊C₈	—	83	2	no	—
199	C₄⋊C₈	—	166	4	no	—
200	C₄⋊C₈	—	166	4	no	—
201	C₄⋊C₈	—	166	2	no	—
202	C₄⋊C₈	—	166	2	no	—
203	#97	—	2	2	no	—
204	#97	—	83	8	no	—
205	#97	—	166	4	no	—
206	C₄×C₈	—	2	2	no	—
207	C₄×C₈	—	166	4	no	—
208	C₄×C₈	C₄×D₈₃	1	2	yes	—
209	C₄×C₈	C₄×D₈₃	1	2	yes	—
210	C₂×Q₈	—	332	2	no	—
211	C₂×Q₈	—	83	2	no	—
212	C₂×Q₈	—	83	2	no	—
213	C₂²×C₄	—	83	2	no	—
214	C₂²×C₄	—	83	2	no	—
215	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
216	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
217	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	166	4	no	—
222	M₁₆	—	166	4	no	—
223	M₁₆	C₂×Dic₁₆₆	1	4	yes	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	83	4	no	—
229	C₂×C₈	—	166	4	no	—
230	C₂×C₈	—	166	2	no	—
231	C₂×C₈	—	166	2	no	—
232	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
234	C₂×C₈	C₂×Dic₁₆₆	1	2	yes	—
235	C₂×C₈	#13050590	1	2	yes	—
236	C₄⋊C₄	—	83	4	no	—
237	C₄⋊C₄	—	83	4	no	—
238	C₄⋊C₄	—	83	2	no	—
239	C₄⋊C₄	—	83	2	no	—
240	C₄²	—	83	4	no	—
241	C₄²	—	83	4	no	—
242	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
243	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
244	C₄²	D₄×D₈₃	1	1	yes	characteristic
245	C₄²	D₄×D₈₃	1	1	yes	characteristic
246	C₂³	C₂×Q₈×D₈₃	1	1	yes	characteristic
247	Q₈	—	332	2	no	—
248	Q₈	—	166	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	332	2	no	—
252	C₂×C₄	—	332	2	no	—
253	C₂×C₄	—	83	4	no	—
254	C₂×C₄	—	83	4	no	—
255	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
256	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
257	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
258	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
260	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
261	C₂×C₄	C₂²×Dic₁₆₆	1	1	yes	characteristic
262	C₂×C₄	C₂×#13050590	1	1	yes	characteristic
263	C₂×C₄	#14581838	1	1	yes	characteristic
264	C₂×C₄	#14581839	1	1	yes	characteristic
265	C₈	—	2	4	no	—
266	C₈	—	2	2	no	—
267	C₈	—	4	2	no	—
268	C₈	—	332	4	no	—
269	C₈	—	166	4	no	—
270	C₈	C₄×Dic₁₆₆	1	4	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	C₂×C₄⋊C₄×D₈₃	1	1	yes	characteristic
274	C₂²	#19128429	1	1	yes	characteristic
275	C₂²	Q₈×Dic₁₆₆	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	166	4	no	—
280	C₄	—	166	4	no	—
281	C₄	#19127748	1	2	yes	—
282	C₄	C₂×C₄×Dic₁₆₆	1	1	yes	characteristic
283	C₄	#19128181	1	1	yes	characteristic
284	C₄	D₄×Dic₁₆₆	1	1	yes	characteristic
285	C₄	#19128372	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	C₄⋊C₄×Dic₁₆₆	1	1	yes	characteristic
289	C₂	C₂×#19127748	1	1	yes	characteristic
290	C₂	#24916071	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₃
Fitting	C₄×#108×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₈₃	C₂²×Q₈
cosocle	C₂²×C₄×C₈₃	C₂⁴
hypercenter	C₄×#108	D₈₃
hypocenter	C₈₃	#22733
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
59. #108×C₈₃	Q₈	semidirect
154. C₈₃	#22733	semidirect
195. #108	Dic₁₆₆	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
M₁₆×Dic₁₆₆	2
C₂×#19127831	2
#24908238	2
#24908237	2
#24907636	2
#24907635	2
#24907371	2
C₄×#108×C₈₃	2
#22733	83